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                    <h1 class="description center-align post-title">Kalma Filter学习笔记</h1>
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                <h1 id="1-前言"><a href="#1-前言" class="headerlink" title="1 前言"></a>1 前言</h1><p>Kalma Filter应用十分广泛，比如导航中</p>
<p>为什么要使用Kalma Filter呢？</p>
<p>因为我们生活中的系统存在许多不确定性：</p>
<ul>
<li>并不存在完美的数学模型</li>
<li>大部分的系统模型不可控，也很难建模</li>
<li>测量传感器自身存在误差</li>
</ul>
<p>如何来理解Kalma Filter呢？</p>
<p><img src="0.jpg" alt="举个栗子！"></p>
<p>让一个人去测量一枚硬币的大小，测量结果为Z_k</p>
<p>k意味第k次测量</p>
<p>假设测量结果为</p>
<p>Z_1 = 50.2mm</p>
<p>Z_2 = 50.1mm</p>
<p>Z_3 = 50.4mm</p>
<p>….</p>
<p>这时我们想要知道硬币真实的数据，只需要把所有数据加起来取平均值就可以了，公式如下，我们将他处理一下：</p>
<p><img src="1.png"></p>
<p>这里我们可以知道，当k越大，1/k越趋近于0</p>
<p><img src="2.png"></p>
<p>也就是说，当k足够大的时候，后续的测试结果就不那么重要了</p>
<p>我们继续处理这个式子变成如下：</p>
<p><img src="3.png" alt="①式"></p>
<p>我们来这么理解：</p>
<p>当前的估计值 = 上一次的估计值 + 系数 * （当前测量值 - 上一次的估计值）</p>
<p>而这个系数K就是卡尔曼增益（因数）</p>
<p>从这个式子里我们可以看到，新的估计值只与上一次的估计只有关，上一次的估计值又只与上上一次的估计值有关，这是一种递归</p>
<p>在卡尔曼中，你不需要很久以前的数据，只需要上一次的就足够了</p>
<p>这里我们引入一个新的概念：</p>
<p>估计误差：E_est  </p>
<p>测量误差：E_mea</p>
<p>E：Error</p>
<p>est：Estimate</p>
<p>mea：Measurement</p>
<p>而K就等于：</p>
<p><img src="4.png"></p>
<p>在k时刻，当E_est_k-1 &gt;&gt;  E_mea_k 时：K→1，①式变为：</p>
<p><img src="5.png"></p>
<p>意味当k-1时的估计误差远大于第k次的测量误差，那么我们的估计值X_k就很趋近于测量值Z_k</p>
<p>同样，在k时刻，当E_est_k-1 &lt;&lt;  E_mea_k 时：K→0，①式变为：</p>
<p><img src="6.png"></p>
<p>意味当我们的估计误差足够大的时候，我们选择更相信测量值</p>
<p>有了上面这些东西，我们就可以着手这个简单的问题了</p>
<ul>
<li>计算：<img src="4.png"></li>
<li>计算：<img src="7.png"></li>
<li>更新：<img src="8.png"></li>
</ul>
<p>有了这三个公式，让我们来试试下面这个问题：</p>
<p>有一个条度为<del>50mm</del>的线，我们想要知道它的确切长度，但幸运的是你的<del>测量误差</del>尺子十分的理想，你觉得它的误差为3mm，但不幸运的是你的<del>估计误差</del>估计很不准，有5mm的误差，这时我们想要知道这条线到底有多长，我们需要结合两个测量</p>
<p>假设我们得出了以下数据：</p>
<ul>
<li>第一次的估计值=40mm</li>
<li>估计误差5mm</li>
<li>第一次的测量值=51mm</li>
<li>测量误差3mm</li>
</ul>
<p>我们可以借助三个公式用Excel表格进行计算，数据输出如下：</p>
<p><img src="9.png"></p>
<p>将公式输入Excel表格并求解画出图像</p>
<p><img src="10.png"></p>
<p>可以看到当我们进行五次迭代以后，数据已经非常接近真实值，至此，前言结束，Excal公式如下，想动手的朋友可以自己去试试：</p>
<ul>
<li>计算部分从第三行开始，所以这里直接放公式</li>
<li>D3=D2+E3*(B3-D2)</li>
<li>E3=F2/(F2+C3)</li>
<li>F3=(1-E3)*F2</li>
</ul>
<h1 id="2-进阶部分"><a href="#2-进阶部分" class="headerlink" title="2 进阶部分"></a>2 进阶部分</h1><p>通过上述栗子我们可以大致了解到Kalman Filter基本思想，但也许有人会说上面的例子里直接数据求算数平均不是更快么？</p>
<p>确实，在一些比较普通的系统中，Kalman Filter的作用并没有那么明显，但在一些复杂系统中，Kalman Filter能发挥出更大的优势</p>
<p>回归正题，我们来看下面一个栗子<del>（数据融合）</del>：</p>
<p>如果我们想要知道一个物体的真实重量，那么我们最简单的办法就是找个秤称一下，但是如果一个称不准呢，那么我们可以再找一个秤，得到两个数据以及两个秤的标准差：</p>
<ul>
<li>Z_1 = 30g</li>
<li>Z_2 = 32g</li>
<li>σ_1 = 2g</li>
<li>σ_2 = 3g</li>
</ul>
<p><img src="11.png" alt="一个标准的正态分布图"></p>
<p>两个正态分布图会有交叉的部分，而如果我们想要知道真实的数据，那么这个真实的数据将会落在交叉的部分，并且会更靠近σ_1那部分</p>
<p><img src="12.png" alt="大概就是这个样子，画的不好见谅"></p>
<p>由此，我们写出一个公式估计真实值Z：</p>
<p><img src="13.png"></p>
<p>K依然是卡尔曼增益，在这里，我们需要求解K，使得方差最小</p>
<p><img src="14.png"></p>
<p>由方差的性质可以将这两个分开</p>
<p><img src="15.png"></p>
<p>将σ_1 =2g和 σ_2=3g带入</p>
<p>可以得到K=0.2</p>
<p>带入公式：</p>
<p><img src="13.png"></p>
<p>得出最优估计值Z=30.4g</p>
<p>可以计算出它的σ = 1.79</p>
<p>再来看一个例子<del>（协方差矩阵）方差协方差在一个矩阵中表现出来</del>：</p>
<p><img src="16.png"></p>
<hr>
<p>状态空间表达：</p>
<p><img src="17.png"></p>
<p>一个简单的栗子，弹簧系统的栗子，F是力，x位移，k弹性系数，B阻尼系数，m质量</p>
<p>得出方程：</p>
<p><img src="18.png"></p>
<p>将上面的公式用矩阵来表示：</p>
<p><img src="19.png"></p>
<p>我们可以将其归纳为：</p>
<p><img src="20.png"></p>
<p>这里的是连续形式，X(t)随时间变化</p>
<p>而我们也可以把它写成离散形式：</p>
<p><img src="21.png"></p>
<p>这里的下标k、k-1、k+1等都是时间单位</p>
<p>但现实世界中存在许多不确定性，我们现有的模型不能很好的反应实际情况，所以我们的模型需要改变一下：</p>
<p><img src="22.png"></p>
<p>这里的w_k-1和v_k分别代表过程噪音和测量噪音</p>
<p>一式计算结果</p>
<p>二试测量结果</p>
<p>这就是状态空间方程</p>
<hr>
<p>虽然过程噪声不可测量，但是我们可以用正态分布来表示</p>
<ul>
<li><p>P(w)~(0,Q)0期望，Q协方差矩阵</p>
</li>
<li><p>Q=E[w,w^T]</p>
</li>
<li><p><img src="23.png"></p>
</li>
</ul>
<p>同理，P(v)~(0,R)0期望，R协方差矩阵，推理过程一样，这里就不写了</p>
<hr>
<p>用数据融合的思想来处理一下我们的公式</p>
<p>先验：</p>
<p><img src="35.png"></p>
<p>后验：</p>
<p><img src="24.png"></p>
<p>X-_k为先验估计，K与误差大小息息相关K属于[0,H-]，为了量化误差，我们引入E_k</p>
<p><img src="25.png"></p>
<p>这里的x_k是真实值，X_k是估计值，化简的出：</p>
<p><img src="26.png"></p>
<p>将结果带入原式：</p>
<p><img src="27.png"></p>
<p><img src="28.png"></p>
<p>当测量结果误差小的时候，K_k=H-，当测量误差很大的时候，K_k=0，至此我们已经推导出卡尔曼增益系数K</p>
<hr>
<p><img src="29.png"></p>
<p><img src="30.png"></p>
<p>有了这个式子，我们就可利用卡尔曼滤波器来预测状态变量的值了</p>
<p>如此一来，我们总结一下我们所得到的式子将上面的X均带上帽子：</p>
<p><img src="31.png"></p>
<p><img src="32.png"></p>
<p>这就是卡尔曼滤波器的全部五个公式了</p>
<h1 id="3-示例"><a href="#3-示例" class="headerlink" title="3 示例"></a>3 示例</h1><p><img src="33.png"></p>
<p><img src="34.png"></p>
<p>大家可以自行使用这个例子代入合适的数据去计算结果，其中除手写部分均为Math Type输入截图，总结不易希望大家喜欢。</p>
<p>参考：<a target="_blank" rel="noopener" href="https://www.bilibili.com/video/BV1ez4y1X7eR/?spm_id_from=333.788.videocard.2">https://www.bilibili.com/video/BV1ez4y1X7eR/?spm_id_from=333.788.videocard.2</a></p>
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